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and 
s= j 2 (see art, 39) (109) 
3 ■ sin 
is the intrinsic equation of the evolute. 
(4) Let the given curve be the equilateral hyperbola, we have 
y(<p) = av/ 2 cot <p ; 
the evolute is s—a (cosec 2<p) ? (110) 
The next five examples are illustrations of art. 43. 
(5) Let the curve be v=e Au * > , its evolute will be 
v=e Bin ®(cot <p-)-cos <p) (HI) 
(6) The tangential equations of the successive evolufes of the curve v—a cos <p are 
a cos 2<p 
Vi sin <p ’ 
v 2 = — 4 a cos <p= — iv ; 
and in general 
».=±4-> (112) 
>W.=±4“>.> (113) 
where the sign + or — is to be used according as m is even or odd. 
(7) Find the evolute of the logarithmic curve. 
The Cartesian equation of the curve is y=e cx , 
and the tangential is v=a log tan <p ; 
and therefore the tangential equation of its evolute is 
v = a cot <p)log tan <p-f- sec 2 <p [. ... ... (114) 
(8) Let the curve be the polar one, g m =a m sinm<p. 
The tangential equation is 
( . mtp 
v=a l sm ^Tif cosec <p, 
and the evolute is 
’‘= ,cot {^l) (U5) 
This result could be easily obtained geometrically. 
(9) The tangential equation of the evolute of the curve 
v=£tan"<p 
is 
v y =v{{n-\-\)cot <p+wtan <pj- (118) 
Hence the tangential equation of the evolute of the common parabola is 
v { =v(2 cot<p-f tan <p). 
(117) 
